## Kategorie: ‘Research Topics’

## Bayesian Model Selection for Complex Shallow Flow

*Prof. Dr. Julia Kowalski*

Shallow flow models are applied to a wide range of science and engineering fields including open channel hydraulics, weather forecasting, landslides in geo-hazard engineering, granular transport in chemical engineering, or coating processes in production engineering. Shallow flows have a much smaller height than length, which justifies depth-averaging yet comes at the price of losing vertical information. Many shallow flow flavors have been proposed in the past. However, it often is not obvious, which model candidate would be the best for a certain situation. In this project, we develop a Bayesian model selection approach to infer on the most plausible candidate process model given a collection of observational data. We will furthermore integrate goal-oriented Gaussian process emulation to increase computational feasibility.

The research goals of this project are:

- Investigate Bayesian model selection approaches for complex shallow flow
- Integrate goal-oriented Gaussian process emulation to increase computational feasibility
- Investigate the impact of different observation data portfolios on selection result

## Multi-Scale Modeling, Model-Order Reduction and Uncertainty Quantification for Transpiration Cooling

*Prof. Michael Herty*

In this project we focus on constrained Bayesian inversion for problems governed by multi–scale partial differential equations, particularly for applications in transpiration cooling. A direct resolution of the inverse problem using existing approaches is prohibitive due to the need to resolve different scales numerically. We build upon the PIs‘ expertise on Bayesian inversion and machine learning methods for large-scale systems, asymptotic analysis for effective models and generalized polynomial chaos expansion (gPC) and optimization to develop novel constrained Bayesian inversion methods for multi–scale inverse problems.

We are interested in the analysis and development of optimization strategies for multi–scale fluid flow models. To obtain a successful method several challenges have to be addressed:

1) fast and reliable surrogate model to address the multi–scale aspect,

2) quantification of the propagation of uncertainty in model hierarchy and in the inverse problem and

3) development of a Bayesian framework under constraints.

## A Hierarchical Framework for Bayesian Optimal Experimental Design

The project will advance the state of the art in hierarchical non-intrusive UQ techniques, such as multi-level and multi-index methods, for goal-oriented optimal experimental design within complex models with high dimensional inputs.

Experimental design is an essential topic in engineering and science. Acquiring relevant information about processes and environments is paramount nowadays. Such cases include, among others, recording weather patterns, measuring traffic density, and tracking industrial process parameters. These data once obtained, provide input for predictive modeling and process optimization. Experiments are meant to provide meaningful information about selected quantities of interest. An experiment may assume different set-ups in a broad sense and can be time consuming or expensive to perform. Therefore, the design of experiments plays a vital role in improving the information gain of the experiment.

Experimental design allows us to optimize the locations of sensors to achieve the best estimates and minimize uncertainties, especially for real, noisy measurements. For instance, determining exactly how many sensors (shown in the figures) to use and their optimal location has significant implications for the reliability and value of the information obtained and for the cost of the measurement system itself. Thus, finding the best set-up for the design of experiments is the main concern of Optimal Experimental Design (OED). In Bayesian OED, one attempts to optimize the experimental set-ups so that the sensitivities between the unknown model parameters and the measurements are maximized.

Our goals in this project are:

Development of advanced hierarchical uncertainty quantification (UQ) techniques for complex systems;

Construction of the corresponding hierarchical Bayesian Optimal Experimental Design techniques for both static and dynamic inverse problems;

Develop novel stochastic optimization approaches tailored to the Bayesian Optimal Experimental Design setting.

## Model-Controlled Bayesian Inversion for Geophysical Inverse Problems

*Prof. Florian Wellmann, Ph.D.*

This sub-project be advised jointly by Prof. Florian Wellmann at RWTH Aachen and Prof. Omar Ghattas at UT Austin. In the context of inverse problems, the field of geosciences provides formidable challenges: the parameter space is large, the applied mathematical models are highly complex, and the available data of diverse quality. We seek here an outstanding candidate to investigate novel geological modeling approaches to address challenging geophysical inverse problems in subsurface applications, for example groundwater and geothermal exploration.

## Model Order Reduction for Goal-Oriented Bayesian Inversion of with High-Dimensional Parameter Spaces

*Prof. Karen Veroy-Grepl, Ph.D.*

In this project, we focus on goal-oriented Bayesian inversion of problems governed by partial differential equations, particularly for applications with high-dimensional parameter spaces. The solution of the discretized inverse problem is often prohibitive due to the need to solve the forward problem numerous times. We thus build upon our expertise on Bayesian inversion for large-scale systems and model order reduction to investigate the use of model order reduction methods to accelerate the solution of Bayesian inverse problems. We intend to use the reduced-basis method and trust region methods to reduce the computational cost in problems with high-dimensional parameter spaces.

## Nonlinear Reconstruction for Probability Densities in Rarefied Gas Flow

*Prof. Dr. Manuel Torrilhon*

Rarefied gas flows, or equivalently flows in microscopic settings, are one of the most prominent examples of flow processes where classical models of thermodynamics fail and enhanced non-equilibrium models become mandatory for simulations. This requires to consider the probability density of the particle velocities in the context of kinetic gas theory an Boltzmann equation. This project has several aims.

There is need to develop a hybrid reconstruction technique for the probability density combining the context of moment models, discrete velocity schemes and direct simulation Monte-Carlo in order to switch between the methods whatever is optimal. Another related challenge is the efficient evaluation of the Boltzmann collision operator. These aspects must be combined in numerical experiments based on the new technique for near-vacuum and supersonic flows.

## Constitutive Reconstruction for Evolving Surfaces

*Prof. Roger A. Sauer, Ph.D.*

The aim of this research project is to develop a computational framework for the reconstruction of material models for deforming and evolving surfaces.

This is done in the framework of nonlinear finite elements and then applied to challenging problems from science and technology.

The work will build on recent work on the reconstruction of loads acting on deforming shells (see figure).

The collaboration partners at UT Austin are Chad Landis and Thomas J.R Hughes.

## Methods for Demand-Side-Management in Process and Chemical Industry

*Prof. Alexander Mitsos, Ph.D.*

The shift from fossil-based to renewable energy sources brings with it fluctuation of energy supply. At the same time the energy demand also has strong time variation. As a consequence it is not sufficient to provide energy, but rather it is necessary to match the times of supply and demand. Options to do so include dispatchable power sources, energy storage and demand-side management. The process industry consumes large amount of energy and thus demand-side management in that industry can substantially contribute to solving the time variability.

However, to be able to ensure such a shift in the process industry there is the need for new computational methods. In particular it has been recognized that merging of on the one hand design&operation and on the other scheduling&control is required. These subdisciplines of PSE are traditionally considered separately. Moreover, handling uncertainty of both demand and supply adds a big challenge.

Both AVT.SVT in RWTH and the labs of Baldea/Edgar at UT have been among the first groups to consider this challenge and propose both methodologies and novel technological options. The purpose of the common project is to combine our methodologies and in collaboration with other CES groups overcome the computational challenges associated.

## Metric-Based Anisotropic Adaptation for Optimal Petrov-Galerkin Methods

*Prof. Georg May, Ph.D.*

Certain Petrov-Galerkin methods (*optimal *PG schemes) deliver inherently stable formulations of variational problems on a given mesh, by selecting appropriate pairs of trial and test spaces. Depending on the choice of norms, quasi-optimal and robust schemes, e.g. for singularly perturbed problems, with stability constants independent of the perturbation parameter, may be obtained.

Optimal PG schemes can be interpreted as minimal residual methods, minimizing the residual in the dual norm (of the test space). In fact, some schemes compute the Riesz representation of the dual residual as part of the solution methodology. In this sense, these optimal Petrov-Galerkin schemes come with a built-in error estimate. Adaptation has therefore been traditionally strongly linked with optimal PG schemes.

Conventionally, adaptation means locally refining the given discrete mesh. A step forward would be the global optimization of the mesh with respect to the inherent error information coming from the optimal Petrov-Galerkin framework. In this context, metric-based continuous mesh models are ideal candidates. A metric-conforming mesh is a triangulation whose elements are (nearly) equilateral under the Riemannian metric induced the continuous mesh model. (Analytic) optimization techniques, based on calculus of variations, may be applied to the metric field, rather than the discrete mesh. The task to produce a mesh is thus delayed until an optimized metric is available.

The error models driving previous continuous-mesh optimization methods need to be adapted to the error representation coming directly from the optimal PG methodology. The goal is to formulate the correct continuous-mesh error models for the optimal Petrov-Galerkin methodology. Nontrivial applications, such as highly convection-dominated problems will be considered to validate the concept.

## Boundary Conforming Smooth Spline Spaces for Isogeometric Analysis

*Prof. Dr. Leif Kobbelt*

The generation, adaptation, and modification of digital 3D models is an

essential prerequisite for many steps in a simulation task. Hence it is of fundamental importance to have efficient algorithms that automatically optimize geometry representations for a given set of requirements. The typical simulation workflow is based on a number of different geometry representations that are considered most suitable for the various stages i.e. NURBS B-rep representations for design or polygon meshes for numerical analysis. Consequently, data conversion steps are necessary which often make the overall process inefficient and error-prone.